Skew polynomial rings, Gröbner bases and the letterplace embedding of the free associative algebra

In this paper we introduce an algebra embedding ι:K〈X〉→S from the free associative algebra K〈X〉 generated by a finite or countable set X into the skew monoid ring S=P⁎Σ defined by the commutative polynomial ring P=K[X×N⁎] and by the monoid Σ=〈σ〉 generated by a suitable endomorphism σ:P→P. If P=K[X] is any ring of polynomials in a countable set of commuting variables, we present also a general Gröbner bases theory for graded two-sided ideals of the graded algebra S=⊕iSi with Si=Pσi and σ:P→P an abstract endomorphism satisfying compatibility conditions with ordering and divisibility of the monomials of P. Moreover, using a suitable grading for the algebra P compatible with the action of Σ, we obtain a bijective correspondence, preserving Gröbner bases, between graded Σ-invariant ideals of P and a class of graded two-sided ideals of S. By means of the embedding ι this results in the unification, in the graded case, of the Gröbner bases theories for commutative and non-commutative polynomial rings. Finally, since the ring of ordinary difference polynomials P=K[X×N] fits the proposed theory one obtains that, with respect to a suitable grading, the Gröbner bases of finitely generated graded ordinary difference ideals can be computed also in the operators ring S and in a finite number of steps up to some fixed degree.